### Abstract

The incompressible Stokes equations can classically be recast in a boundary integral (BI) representation, which provides a general method to solve low-Reynolds-number problems analytically and computationally. Alternatively, one can solve the Stokes equations by using an appropriate distribution of flow singularities of the right strength within the boundary, a method that is particularly useful to describe the dynamics of long slender objects for which the numerical implementation of the BI representation becomes cumbersome. While the BI approach is a mathematical consequence of the Stokes equations, the singularity method involves making judicious guesses that can only be justified a posteriori. In this paper, we use matched asymptotic expansions to derive an algebraically accurate slender-body theory directly from the BI representation able to handle arbitrary surface velocities and surface tractions. This expansion procedure leads to sets of uncoupled linear equations and to a single one-dimensional integral equation identical to that derived by Keller & Rubinow (J. Fluid Mech., vol. 75, 1976, p. 705) and Johnson (J. Fluid Mech., vol. 99, 1979, p. 411) using the singularity method. Hence, we show that it is a mathematical consequence of the BI approach that the leading-order flow around a slender body can be represented using a distribution of singularities along its centreline. Furthermore, when derived from either the single-layer or the double-layer modified BI representation, general slender solutions are only possible in certain types of flow, in accordance with the limitations of these representations.

Original language | English |
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Article number | R1 |

Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Journal of Fluid Mechanics |

Volume | 850 |

DOIs | |

Publication status | Published - Sep 2018 |

Externally published | Yes |

### Keywords

- boundary integral methods
- low-Reynolds-number flows
- slender-body theory

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## Cite this

*Journal of Fluid Mechanics*,

*850*, 1-12. [R1]. https://doi.org/10.1017/jfm.2018.483